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The problem presented involves a two-column proof for the **Alternate Interior Angles Theorem**. The given information states that lines \( c \parallel d \) and you are tasked with proving that \( \angle 4 \cong \angle 6 \).

Let's break this down by focusing on the statements provided in the table:

1. **Statement 1: \( c \parallel d \)** – This is given information.
2. **Statement 2: \( m\angle 6 + m\angle 7 = 180^\circ \)** – This statement refers to the **Linear Pair Postulate** or **Supplementary Angles**, which is true because \( \angle 6 \) and \( \angle 7 \) form a straight line (linear pair) along the transversal cutting through parallel lines.
3. **Statement 3: \( m\angle 4 + m\angle 7 = 180^\circ \)** – This involves the same reasoning as Statement 2, because \( \angle 4 \) and \( \angle 7 \) are also a linear pair.

Now, for the missing reason in Statement 3, the justification should be similar to that in Statement 2, which refers to **Same Side Interior Angles are Supplementary**. Since the two angles add up to \( 180^\circ \), it confirms this relationship.

Therefore, the correct reason for Statement 3 should be:

**"Same side interior angles are supplementary."**

Would you like further details on how this proof works or how the Alternate Interior Angles Theorem is applied?

Here are some related questions for further understanding:

1. What is the definition of alternate interior angles?
2. Why do alternate interior angles become congruent when lines are parallel?
3. How does the Linear Pair Postulate apply to angles formed by transversals?
4. Can the same reasoning be applied to other angles in this diagram? Which ones?
5. What are some other angle relationships in parallel lines cut by a transversal?

**Tip:** In geometry, understanding angle relationships formed by parallel lines and transversals is crucial for solving many types of proofs and problems.

A portion of the two-column proof for Alternate Interior Angles Theorem is shown. Given: c parallel to d. Prove: angle 4 is congruent to angle 6. Missing the reason for Statement 3: m(angle 4) + m(angle 7) = 180.

Learn how to complete a two-column proof for the Alternate Interior Angles Theorem. Given parallel lines cut by a transversal, this problem focuses on proving that angle 4 is congruent to angle 6 using the Same-Side Interior Angles Theorem. This is a key geometry concept for students in Grades 8-10.

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